and
AH_[2]
to itself defined by f(1) = 6.
AH_[2]
to 
defined by
f((1,0)) = (3,-5) and f((0, 1)) = (1, 4). (As usual, I will often write
column vectors as row vectors -- you are expected to interpret them
correctly from the context.)
AH_[2]
|
|
|
|
to 
is defined
by the matrix 
where the
matrix was defined with respect to the basis 
(in this order)
for the codomain and 
(in this order) for the domain.
Suppose the linear map g from 
to 
is defined
by the matrix 
where the
matrix was defined with respect to the basis 
(in this order)
for the codomain and 
(in this order) for the domain. Let
h = f(g) be the composition of the two linear maps.
AH_[2]
AC_[4]
AC_[4]
and 
AC_[4]
AC_[4]
.
for the codomain
and 
for the domain is:
|
|
(in this order) is
|
and 
be linear maps of finite
dimensional vector spaces. Let 
, 
, and
be bases of U, V, and W respectively. Let 
and 
be
the matrices of f and g respectively, where each matrix is written using
the above bases (in the order specified). Let h = f(g).
AH_[2]
.
.
.
.
to a matrix is equivalent to
multiplying it on the left by the matrix 
for i = 1, 2, ..., k.
Then the result of applying the operations 
(in this
order) to the matrix is equivalent to multiplying the matrix on the left
by the product 
.
to a matrix is equivalent to
multiplying it on the left by the matrix 
for i = 1, 2, ..., k.
Then the result of applying the operations 
(in this
order) to the matrix is equivalent to multiplying the matrix on the left
by the product 
.
.
(Enter your answer as exact fractions, e.g. 4/3 rather than a decimal
approximation.)
AH_[2]
|
to itself defined by the
matrix 
where we are
using the standard basis. Consider the basis 
.
AH_[2]
|
, then using the results of the
last exercise, one can calculate 
to be exactly
|
.
AH_[2]